Appl. Sci. Res. 25 February 1972

REFLECTION AND TRANSMISSION OF ELASTIC WAVES BY A MOVING SLAB

D. C E N S O R

Department of Environmental Sciences, Tel-Aviv University,

Ramat Aviv, ISRAEL

J. A B O U D I a n d D. N E U L A N D E R

Department of Engineering Sciences, Tel-Aviv University,

Ramat Aviv, ISRAEL

Abstract

Scattering of elastic waves by a moving slab is considered. Two cases corre- sponding to strong contact and good lubrication are treated. I t is shown that the motion introduces new effective compressional and shear wave velocities in the moving slab. The amplitude of the reflection and transmission coef- ficients are given for various angles of incidence, frequencies and velocities of motion.

Nomenclature

A , B e, [ k , K Ptj, Eij t t t

v

X, y , z

F

2 , # p 4,,4, go

amplitude coefficients angles of propogation of compressional and shear waves, respectively propagation vectors stress and strain tensor components, respectively time displacement vector velocity of moving medium cartesian coordinates compressional and shear wave speeds, respectively inertial frame of reference Kronecker's delta Lam~'s constants density potentials angular frequency

-- 313 --

314 D. CENSOR, J. ABOUDI AND D. NEULANDER

subscripts p, s refer to cornpressional and shear parameters, respectively r , t , + , - refer to waves: reflected, transmitted, forward inside the slab,

backward inside the slab, respectively

§ 1. Introduction

The interaction of elastic waves with moving systems is of interest, both from theoretical and practical point of view. The acoustical case, which constitutes a special case of the following problem, has been treated extensively.

The scattering of acoustic waves by uniformly moving media is discussed by Morse and Ingard [1]. Keller [2] considers the case of scattering by a moving half space. See also Miles E3], and Ribner [4]. Yeh [5] considers a moving fluid layer, and E6J, scattering by a moving fluid cylinder.

Some engineering applications, e.g., reflection of sound waves from jets, are indicated by the above mentioned authors. From the theoretical point of view, it is desirable to understand the behaviour of elastic waves in similar configurations, because of the transversal (shear) waves encountered here, In particular, the interaction of the compressional and shear waves is a new feature. By letting the shear wave velocity vanish, in one or more regions, special cases of mixed acoustical-elastic or acoustical situations are obtained. The transmission of elastic waves through sliding objects is relevant to machine noise and measurement of parameters (e.g., velocity) of moving parts which are not otherwise accessible. In special cases of elastic objects moving in fluids, the problem might be relevant to underwater acoustics. Conversely the motion of a fluid in an elastic duct might be sounded by means of the scattered waves.

The correct choice of the boundary conditions is a severe problem, depending on tile practical situation in question. In many cases there is good lubrication between the moving parts. It is plausible to idealize this situation by assuming that tangential stresses and displacements will not be transmitted. In other cases strong friction may exist so that these components will be practically transmitted across the boundary. In this paper both limiting cases are considered.

The geometry chosen here is of an elastic slab uniformly moving parallel to its surface. We will show that two new effective wave

ELASTIC WAVE PROPAGATION 315

velocities are encountered here, which reduce to the conventional wave velocities for the slab at rest. To study the intrinsic velocity effects, computations were performed, for the case where the slab and the surrounding medium have the same parameters in their proper frames of reference. Reflected and transmitted waves are present, and for certain velocities and angles of incidence criticality and resonance effects occur.

§ 2. Statement of the problem

Let us consider the problem of scattering of plane, space and time- harmonic elastic compressional (p) waves, by a slab moving parallel to its surfaces with a constant velocity v = v2, as observed from tile frame of reference of the laboratory, in which the surrounding medium is at rest. See Fig. 1.

The incident compressional wave is of the form

c~O =- e i k p ' r - i ~ t , k p . r =- z c o s e @ x s i n e.

¢o = o , = (1)

In (1) 40 and ¢0 are the displacement potentials as derived subsequently; kp----o)/~1 is the propagation constant in the ex- ternal medium, where ~1 is the compressional wave velocity in the external medium, designated 1. Subsequently the time factor e i~t is suppressed throughout.

Two cases are chosen for the boundary conditions at z----- -+-d as explained above: a) continuity of displacements and stresses, and b) vanishing tangential stresses.

For both cases the problem is consistent and determinate, however, these are not the only ways to pose the problem. For example, since the contact is not perfect, it can be argued that only a fraction of the tangential displacements and stresses are trans- mitted across the boundary. Here, however, these proportionality factors are considered as unity. As long as the exact nature of the contact is not well understood, these models must be considered to be heuristic.

To satisfy the above boundary conditions similarly to the case of media at rest, reflected and transmitted compressional and shear (S) waves are required,

316 D. CENSOR, J. ABOUDIAND D. NEULANDER

Cr ~--- A r e ik'~'r, ~pr ' r : - - z o o s er -~- x s i n er,

6r = Br e ik' 'r, f~sr'r = - -z c o s / r + x s in / r ,

Ct = A t e ik~''r, /~pt'r : z c o s e t + x s i n e t ,

St : Bt e ik°''r, /~st" r : z c o s / t + x s in / t ,

4)+ -~ A+ e i~+'r, /~p+. r = z cos e+ + x sin e+,

~b+ = B+ e i~S+'r, /~s+" r = z co s /+ + x s in /+ ,

4 - = A_ e iKp .r, /~p_. r = - -z cos e_ + x sin e_,

6 - = B - e i . . . . ~, / ( s - ' r = - - z cos /_ + x sin /_, (2)

where K refers to effective propagat ion constants in the inter ior of the slab, as defined below.

In order to show tha t plane waves as described in (2) can exist in the un i formly moving medium, and to find the re levant new para- meters, a Galilean t rans format ion is applied to the fundamenta l equat ions of elasto-dynamics.

In the f rame of reference F' (x ' , y ' , z' , t') of the medium at rest, we have

p(d /d t ' )2u ' : (~ + # ) V ' J 7 ' . u ' + # V ' 2 u ', (3)

Transforming (3) into the l abora to ry frame of reference P ( x , y , z , t ) yields, for infinitesimal deformations,

p[(~/~t) + v . V l 2 u = (~ + ~ ) V V . u + ~V2u, (4)

since u : u ' . Defining displacement potent ia ls in the usual way, such t ha t

u = V¢ + V × ¢ , (s)

it follows tha t ¢ ' : ¢, ~ p ' : ~p. For harmonic t ime dependence e -i°)t we obtain,

--o)2(1 - - v . V/io))2¢ = ~2V2¢, a 2 = (2 + 2#)/p,

_ ~ 2 ( 1 _ v . V/io~)2¢i = ~2V2¢i , /~2 = ~/p, (6) i = x, y, z.

Consider a plane wave

(~' : ei~p"r'-io~'t" : ~ ~ ei~p "r- lot ,

t ? Kp : ~o/a, (7)

subs t i tu t ing the Galilean t rans format ion in (7) leads to

ELASTIC WAVE PROPAGATION 317

A ¢ t ~,/o~'= 1 + v.Kp/~, Kp = K;. (8) Defining

~e*f = co/Kp = ~(1 q- V'Rp/O~), (9)

and substituting (7) with (8) into (6), shows that (9) is consistent. A similar discussion leads to

fleer = o) /Ks = ~3(1 + v'$~s/ f i ) . (10)

Thus in F we have new effective wave velocities (9), (10), this implies, formally, new Lame' constants for the moving medium,

Zeff p[~,~ 2 = - - 2fi~fe], (ll)

/*off - - P f i e ~ .

Since the boundary surfaces are stationary in the laboratory system no Doppler frequency shifts are detected, and the frequency o) is preserved as the wave enters the moving medium 2.

Subject to the above argument our problem is to find the coef- fieients defined in (2).

§ 3. Solut ion of the bounda ry va lue problem

Case a. The boundary conditions for the continuity of the displace- ments and stresses are:

Uxl ~--- ~Ax2

Uzl ~ ~z2 at z = 4-d, (12a)

~Pzxl = ])ZX2

P=i = p=~

Pfj = 2V" ud i j -4- 2/*eij,

e i j = [(o/~xj)ui + (O/~xduj]/2,

}1, i=/ d~j= 0, i : # i .

Case b. The boundary conditions for the continuity of the normal displacements and stresses, and vanishing tangential stresses, are:

~z l 7_ Nz2 ]

Pzx l = Pzx2 = O i a t z = ~=d. (12b)

P=I = P=2

318 D. CENSOR, J . ABOUDI AND D. N E U L A N D E R

1 J I

X

~ Z

2 d

2 I

Fig. 1. Geometry of the problem. A compressional wave, denoted by P, is incident at an angle e on an elastic slab of width 2d. The surrounding medium and the medium of the slab are denoted by 1,2 respectively. The slab is uniform!y moving in the x direction with velocity v Compressional, shear waves are denoted by P, S and propagating in directions e, / (or e', / ' in the

internal domain) respectively.

This implies Snell 's law,

sin e sin er s i n / r

C~l ~1 f l l

s in /+

fl2eff

sin e+ sin e_

C~2eff O~2eff

s i n / - sin et s i n / t ( 1 3 )

Subjec t to Snell 's law, the angles

e - - e r = e t , l = / r = l t , e' = e + = e - , t ' = 1 + = 1 - ,

See Fig. 1. Express ing the d i sp lacement u and the Pij in t e rms of the

po ten t ia l s ¢, ~ and subs t i tu t ing in (12), we ob ta in eight equat ions for the eight unknown coefficients in (2).

The vec tor t e r m in case a is represen ted in case b b y 5I.

I

o .4

% :£ o o i l o o

i l S . s l I

o

s ¢ > , s

~, ® 'S

, o s I ]

- -~ X

~ - a o o .< _< o I

s~ a ? 'a s £

t'~S I o o :~ I I o

i

£

i 5 ÷

4

W v

. ~ X . ~ X

X

e X ~ ~

'~w " ~

~× ×

£

I i

s o ~ ~

~:~ s~ ~ i o o £ o o

I

o I X

x

i

'm 8 o o o I o o ~ ~_

%

8

~ .~ ÷

o o o I o o I ,~

.~ ~ ~ x~ ~-. ~ ~. "~ ~

o '~ I o ~

% x

, . ~. + @

o o ~ I ~ ~ o I

- ~ , ~ - = *

o I ~T '~ ~ --.-.-t o I "~ x

o %

• ~, ~ "m ÷ ~ "~

o I I o I ~ o I

,.~ "~ o o o ,~ o o

+

. ~ I o o o I o o i

v

J

,..¢

.?

b, t~ + -m ,..,2 "~ i,~, I 0 0 0 ~ 0 0 I

II F' "1

X

E L A S T I C W A V E P R O P A G A T I O N 321

§ 4. Discuss ion

In order to bring out the intrinsic veloci ty effects the two media 1,2 are t aken identical in their proper frames of reference, #z = #2,

21 = ~2, p1 = p2.

From Snell's law (13) we see that the direction of the compression- al wave emerging from the slab is identical with the direction of the incident wave. This however, does not imply recirpocity, since the directions of propagat ion with respect to the veloci ty v determine the equivalent parameters . See Morse and Ingard [1], page 710, for the case of a moving half space. This means tha t in Fig. 1 we m a y inver t the direction of the incident and t r ansmi t t ed P waves, bu t the waves inside the slab will be different.

Snell's law (13) gives the angles for which cri t ical i ty takes place because of the veloci ty effect. For a given angle of incidence e, let us invest igate the behaviour of the reflected, t r ansmi t t ed and re- f rac ted waves, as a funct ion of the veloci ty v, according to

sin e' : sin e/(1 - - v sin e), ~1 = c~2 = 1, (15)

which follows from (9) and (13), and similarly for s in / ' . For - - c o < v < Vo, where v0 = cosec (e) - - 1, there is no criticality. At v - - v0 according to (15) sin e' -- 1, i.e., e' = 7:/2 so tha t v > vo,

a propagat ing P wave cannot pene t ra te into the slab. See Fig. 2. For v 0 < v < v 0 q - 1 we have s i n e ' > 1, which implies e ' =

s in e ' /;

c e c e - I I C ° s e c e l c o s e c e * l ~ v

i .,'5 - I . . . . . . i . . . . . . I

/i I I I I ,

I f E

[ q i

=ef t I

- -JBef f I

[ COSEC e 2 COSec •

. J . . . . [ X [

I = o f f

( a ) (b l

F i g . 2. S k e t c h of (a) s i n e ' a n d (b), t h e e f f e c t i v e v e l o c i t i e s , a s a f u n c t i o n

of v. T h e c o m p r e s s i o n a l w a v e v e l o c i t i e s a r e c h o s e n a s ~1 = ~2 = 1, a n d

t h e s h e a r w a v e v e l o c i t i e s a r e //1 = /~2 = l / x / 3 .

322 D. C E N S O R , J . A B O U D I A N D D . N E U L A N D E R

= i e " 4 - r ~ / 2 where e" is reai and negative to ensure an ex- ponent ia l ly decaying wave with increasing z inside the slab. This is consistent for the reflected wave inside the slab too.

For v0 q- 1 < v < v0 -~ 2 we have - - o o < sin e' < - - i , there- fore e ' = i e " - - ~ / 2 where e" is real and negative. This again ensures a t tenuated waves, taking into account the fact that the effective wave velocities in this range are negative. See Fig. 2.

For v 0 - [ - 2 < v < o o we have - -1 < s i n e ' < 0 , i.e., - - ~ / 2 < < e' < 0. In this range there is no criticality, the refracted waves have real directions of propagation. Note that for v > v0 -k 1, the effective velocities are negative. This takes place at supersonic velocities v > ~2 1. The same arguments apply to s in / ' . Note that O~eff and /~eff become infinite at the same veloc i ty v0 -~ 1, see Fig. 2.

In Fig. 3 the amplitude of the reflection and transmission coef- ficients are given for case a for angles of incidence in the range 0 to

1.0

O8

O6

0.4

0.2

O0

0

.,.'" v=.l.O - - v~0.5 . . . .

v , l.S .........

¢"' . . . . . i / i 06~ ................ ) ,~.\ ; ',,/" / / / . . . . . . o.~L . /I .... ,....X\

' ' go 8'0 , ; o' ~o ' ' ' ' ~e 60 80 IO0 20 40 e

IO

O0 ~ k.." ~.__~

F i g . 3 . Absolute value of reflection and transmiss ion coefficients for case a , a s a f u n c t i o n o f a n g l e o5 incidence. For v = - - 1 there is no criticality. For v = 0 . 5 criticality takes place f o r P and S waves at 42 degrees, 69 degrees, respectively. For v = 0 . 8 - 3 4 degrees, 47 degrees. For v = 1 . 5 - 24

degrees, 29 degrees.

ELASTIC WAVE PROPAGATION 323

90 degrees, at various velocities and a constant frequency m = 1. As predicted by the theory no criticality occurs for negative

velocities, and the curves are smooth. For positive velocities criti- cality occurs for P and S waves, producing abrupt jumps in the curves at those special angles. Compared to the problem of a moving elastic half-space the present situation is further complicated because of resonance effects. I.e., for certain angles of incidence the waves inside the slab interfere constructively or destructively. This is seen in Fig. 5 too.

At normal incidence e = 0 degrees, according to (9), (i0) the

LO

08

0.6

1o.4

0.2

(3-O

~o,1 I ~ 15 d e ~ l ' e e s _ _ . _ 0.8 ~' = (,o de2tees ........

, . . . . . . . _°° .F.72\ \ " i ~ ~ i f f ~ f 04 ,. / " \ "

-~'o 20 o'o Io '.0 '.o - - v ' -,!o o'o ,' 2'o ,~ -2.O ~ v

10

08

0.6

t °4

0.2

0 D

f

o.4 ~ ....... fx ,~',, j~ ,,~

oo h ~ X , j i \d

Fig. 4. A b s o l u t e va lue of re f lec t ion a n d t r a n s m i s s i o n coeff ic ients for case a,

as a f u n c t i o n of ve loci ty , for v a r i o u s angles of incidence. T h e effect ive

veloci t ies t e n d to i n f in i t y a t t he fo l lowing p o i n t s ;

e -- 15 degrees , vo~ = 3.8; e - - 30 degrees, Voo = 2 ;

e = 45 degrees, vo~ - - 1.41; e = 60 degrees, v~ = 1.15.

F o r e - - 15 degrees c r i t i ca l i ty occurs for P w a v e s a t v = 2.9. F o r e = 30

degrees , c r i t i ca l i ty occurs for P a n d S w a v e s a t v = 1.05, v - 1.45 re-

spec t ive ly . F o r S a n d P w a v e s c r i t i ca l i ty ceases a t v = 2.6 a n d 3.0, re- spec t ive ly . F o r e = 45 degrees c r i t i ca l i ty for P, S w a v e s a p p e a r s a t

v = 0.45, 0.85. Respec t ive ly , a n d v a n i s h e s for S, P w a v e s a t 2.0 a n d 2.45,

r e spec t ive ly . F o r e = 60 degrees c r i t i ca l i ty for P, S w a v e s a p p e a r s a t 0.2,

0.6 r e spec t ive ly , a n d d i s s a p p e a r s for S, P w a v e s a t 1.75, 2 . 2 , r e spec t ive ly .

324 D. CENSOR, J. ABOUDI AND D. NEULANDER

velocity effect vanishes, consequently (for identical media) the reflected waves disappear, the transmitted P wave is of unity magnitude and the S wave vanishes. At a grazing angle e--> 90 degrees there are no transmitted waves, since nothing penetrates into the slab, the reflected wave is a P wave only.

In Fig. 4 the amplitudes of the coefficients are given for case a as a function of the velocity v, in the range - -2 to 3 for various angles of incidence. As expected, for v = 0 all the coefficients vanish except IAt] -- 1, since the slab has no effect. For negative velocities we have again smooth curves because criticality does not occur. For positive velocities abrupt changes are seen at the points where criticality begins, for P and S waves, and where it ends.

At the point 1 = v sin e, according to Fig. 2 the equivalent wave velocities tend to infinity, i.e., we are dealing with a perfectly rigid slab. At these points the transmitted waves vanish.

In Fig. 5 the amplitudes of the coefficients are given for case a as a function of the frequency for various values of v. Except for

~f ::'::::::':" " .................................................. O~ _ /

i I

ID

Q 8

a 6

~ 2

O ~

e=30 degreas

0 .6 v a - t - -

..... -.. v , 0 . 5 . . . . v , 0 . 8 . . . .

d . 4 v = t , 5 . . . . . . . . . .

~ o J 4 0 *

' " , . . .

Fig. 5. Abso lu te va lue of ref lect ion and transmiss ion coeff icients for case a ,

as a funct ion of f requency , for var ious ve loc i t ies and e = 3 0 degrees. For v = 1 . 5 we are wi th in the range of crit ical i ty .

ELASTIC W A V E PROPAGATION 325

0.75

,.~ 0.50

0,2@

0.0

i / A / I I

I I ..")72'., . f \~ , . /' / / / / , # '~ / I

, I .," 1 41 i~

I ! / / / , , ,

oJ-_A

v =-[0 --

v , O . 5 - - - -

v=O,8 - - ' - -

v ' L 5 .........

I I I I I I I I 0 30 60 90 0 30 60 90

~e ~e

,.o , /!

o,0 v , ,

o,o ii i l l , ..... ,

0.0

Fig . 6. A b s o l u t e v a l u e of r e f l e c t i o n a n d t r a n s m i s s i o n coe f f i c i en t s for c a s e b, a s a f u n c t i o n of a n g l e of i nc idence .

0.50 ~ :

~o.a~ " ' -?~ ....

-- 0.0

v = 1.5, which is in the critical range (see Fig. 2), all curves show resonance effects.

For case b, Snell 's law holds as before, and cri t ical i ty occurs at the same points. However , the different boundary condit ions for case b produce new pat te rns for the various scat tered waves. Similarly to case a, for normal incidence (e = 0) Fig. 6 show zero reflection ampli tudes and the slab has no effect. At grazing angles (e -+ 90 degrees) the displacement produced by the incident P wave become tangent ia l to the interface, hence it behaves as a free surface. The wave is ent i re ly reflected as a P wave.

In Fig. 7 the ampli tudes are shown as a funct ion of the velocity. Cont ra ry to case a (Fig. 4), here for v = 0 the effect of the slab does not disappear. This is a consequence of the b o u n d a ry con- ditions. The fact t ha t tangent ia l stresses are not t ransmi t ted , affects the scat ter ing process even in the absence of velocity. For

326 D. CENSOR, j . ABOUDI AND D. NEULANDER

1,0

0.75

~ 0.50

0.25

0.0

-2

i,/i \!

t -I

I I I O . . v I 2

~'__J

i : i

! Q2'~

i!

0.0

I

e = 15 degrees- e = 30 degrees .... e = 45 degrees e = 60 degrees--'-

"*~.. / ; /",,.] '~ 7 / : i i ,

,-~"~: iXq;/ ,' [ h

' I ~ ~ I I I I I

-I 0 I 2 3

0.75

~0.50

0.25

0.0

Fig. 7.

°E= I "i ,v 0 .0

A b s o l u t e va lue of re f lec t ion a n d t r a n s m i s s i o n coeff ic ients for case b,

as a f u n c t i o n of ve loc i ty , fo r v a r i o u s angles of incidence.

normal incidence (e = 0) only, the ' fault ' will have no effect, hence as the velocity changes from zero, the effect of the slab becomes significant, see Fig. 7 for e = 15 degrees.

A c k n o w l e d g e m e n t

The computat ions connected with this paper were performed at the Computat ion Center of the Tel-Aviv University.

Received 21 January 1971

In final form 30 June 1971

ELASTIC W AVE PROPAGATION 327

R E F E R E N C E S

[1] MORSE, P. M. and K. UNo INGARD, Theoretical Acoustics, McGraw-Hill, New York 1968, eh. 11.

[2] KELLER, J. B., The Journal of the Acoustical Society of America 27 (1955) 1044. E3] MILES, J. W., The Journal of the Acoustical Society of America 29 (1957) 226. [4] RIBNER, H. S., The Journal of the Acoustical Society of America 29 (1957) 435. [5] YEH, C., The Journal of the Acoustical Society of America 41 (1967) 817, see also

43 (1963) 1454. [6] YEH, The C.,JournaI of the Acoustical Society of America 44 (1968) 1216.